Acla Asfronautico Vol. 35, No. 8, pp. 489-500, 1995

Pergamon OO!M-5765(95)00006-2 Copyright Q 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0094-5765195 $9.50 + 0.00

OPTIMAL MASS FOR AEROBRAKING TETHERS?

J. M. LONGUSKI, J. PUIG-SUARI, P. TSIOTRAS and S. TRAGESSER

School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-1282, U.S.A.

(Received I.? August 1994)

Abstract-Earlier work has demonstrated the feasibility of using aerobraking tethers for the exploration of the solar system. In fact, compared to chemical propulsion, the tether mass is usually much less than the required propellant mass. The basic concept involves an orbiter and a probe connected by a thin tether. The probe is deployed into the atmosphere of a planet where aerodynamic drag decelerates it. The tension on the tether provides the braking effect on the orbiter, thus eliminating the need for a propulsive maneuver. During the maneuver the orbiter travels outside the atmosphere, and does not require heat shielding. In the previous work a suboptimal solution was found where the system maintained a near vertical orientation during the fly through. In this paper we consider the minimum tether mass required for specified aerocapture conditions. As an intermediate step, we find the trajectory which provides the minimum tension on the tether. The fact that the orbiter must remain outside the atmosphere is introduced as an altitude constraint. The results are significant for future solar system exploration.

1. INTRODUCTION

In the history of the exploration of the solar system, some of the most successful and most ambitious missions have involved dual vehicle spacecraft. In such a mission (the Viking program being the best known) one vehicle (a probe or lander) is delivered to the planet’s surface or atmosphere, while the second one (the orbiter) remains in orbit around the planet. The Galileo spacecraft, currently on its way to Jupiter, is representative of several missions being proposed in this category. When the spacecraft arrives at the target planet, the orbiter performs a propulsive maneuver to achieve capture, while the probe relies on an aerobraking maneuver to decelerate.

The aerobraking tether shown in Fig. 1 eliminates the need for the orbiter propulsive maneuver. The spacecraft consists of an orbiter and a probe that are connected by a thin tether. When the vehicle arrives at the planet, the probe flies into the atmosphere, as before, while the orbiter is decelerated by tether tension, thus eliminating the need for propellant. Note that the orbiter remains outside the atmosphere during the maneuver and does not require additional aerodynamic shielding. After capture has occurred, the tether may be severed, allowing the probe to land on the planet, or the system may remain together and additional aerobraking maneuvers can be performed to finalize the orbit.

tPaper IAF-93-A.2.13 presented at the 44th International Astronautical Congress, Graz, Austria, 16-22 October 1993.

The concept of using tethers for aeroassisted ma- neuvers has appeared in the literature as early as 1986, being mentioned by Carroll [l] and Purvis and Penzo [2]. The idea was previously proposed at the Jet Propulsion Laboratory in 1984 by Sirlin et al.[3]. Despite the early introduction of the aerobraking tether idea and the great interest that tethers in space have received in recent years, work on tethers in an atmosphere has been very limited. Lorenzini et al. [4] analyzed the behavior of a tethered system in the Martian atmosphere, but the spacecraft was main- tained in circular orbit using thrusters. More recently, Warnock and Cochran [5] studied the orbital lifetime of tethered satellites. This is a very complete analysis of orbit decay but no attempt was made to use the drag for maneuvering.

Analysis by Puig-Suari and Longuski [6] involved modeling the tether as a rigid rod with the conjecture that, if aerobraking is feasible with such a simple model, then a more involved study would be war- ranted. This conjecture is supported by the early work of Lorenzini et al. [4], which demonstrates that a flexible tether, subjected to aerodynamic loads, remains relatively straight, at least in a circular orbit. Even though the rigid rod model does not include flexible effects, it retains the essential behavior of the system by taking into account distributed gravita- tional and aerodynamic forces. Using this model, Puig-Suari and Longuski demonstrate the physical feasibility of aerobraking with a tether in [6] and [7].

In [8], Puig-Suari and Longuski deal with the difficult task of finding targeting conditions. Simple analytical models are developed to facilitate the de- termination of the initial conditions required for aerocapture. First, an analytical approximation for

489

490 J. M. Longuski et al

Orbiter

Fig. 1. Aerobraking tether

the change in velocity during the atmospheric portion 01 the trajectory is found in terms of error functions. Next, perturbation techniques are used to determine approximate equations for the motion of the system outside the atmosphere (i.e. with no aerodynamic effects). These two approximate models provide a way to determine a good first guess for initial con- ditions. The precise initial conditions are found iteratively by performing numerical analysis with the (much more complicated) complete rigid rod model. This makes it possible to define and simulate the behavior of many types of tether aerocapture maneuvers.

Longuski et al. [9] compare the performance of the tether system with that of the traditional propulsive maneuver in missions to all the atmosphere-bearing planets in the solar system. The relative performance of the two systems is determined by comparing the mass of the tether and the mass of the propellant required to capture the orbiter. Despite the fact that no attempt is made to determine the minimum tether mass required for the maneuvers, the results show a clear mass advantage in the tethered system.

The results obtained with the rigid rod are very encouraging, but the flexibility of a real tether may have detrimental effects on the performance of the system. In [lo] and [l l] flexible tether models are developed to permit the analysis of more realistic tethered systems. The results obtained with the flex- ible models indicate that the basic behavior of the tethered system is very similar to that observed with the rigid rod model. This validates the earlier conjec- ture that the rigid rod model might serve as a reasonable tool for the preliminary assessment of tether aerobraking problems. In fact (when the proper care is exercised) the rigid rod model is very faithful to the actual behavior so that flexible analysis

is only necessary in the final stages of the design process.

In this paper optimization techniques are used to determine the aerobraking maneuver which mini- mizes the tether mass. The optimum maneuver may be used to increase the already significant mass advantage of the tether aerobraking system over the traditional propulsive maneuvers.

2. AEROBRAKING TETHER DESIGN CONCEPTS AND RESULTS

In [9] aerobraking tether design concepts are developed for the exploration of the solar system. The model used in the analysis assumes a rigid tether, planar motion and an exponential atmosphere. The study includes missions to Venus, Mars, Jupiter, Saturn, Uranus, Neptune, Titan and a Mars-Earth return mission. In this preliminary study, certain specifications are made which ultimately lead to unique designs for braking into a capture orbit about these planets. These specifications include the following.

(1)

(2)

The fly-through maneuver (Fig. 2) is designed so that the inertial spin rate during atmospheric entry, Q,, is equal and oppo- site to the spin rate during exit, R,,,

% = -% (1)

This requirement is called spin matching. The tether length is specified so that the location of the center of pressure with re- spect to the probe, lPS, coincides with that of the center of percussion, Iv. This require- ment is called center matching:

1,s = l,C (2)

Fig. 2. Aerobraking maneuver.

Aerobraking tethers 491

(3)

(4)

(5)

The ballistic coefficient of the probe is equal to the ballistic coefficient of the tether:

rl /(Co, d) = mp I(&, S,) (3)

where r) is the tether linear density, which depends on the tether mass and its dimen- sions, CD, and C, are the drag coefficients of the tether and the probe, respectively, d is the diameter of tether and S, is the frontal area of the probe. In [9] the values assumed for CD, and Cn, are 2 and 1, respectively. This requirement is called aeromatching. The mass of the orbiter, RI,, and the mass of the probe, mp, are both assumed to be equal to 1000 kg:

m,=m,= 1OOOkg (4)

The tether is assumed to be Hercules AS4 graphite [12] with a tensile strength, cU, and a density, p,:

6, = 3.6 GN/m2

pt = 1800 kg/m’ (6)

(6) The planets are assumed to be in circular, co-planar orbits. Arrival conditions are cal- culated by assuming an interplanetary Hohmann transfer. The spacecraft is then captured into a near-parabolic orbit about the target planet so that:

e = 0.9999 (6)

(where e is the eccentricity with respect to the planet).

From conditions (l)-(3), and assuming that the tether is nearly vertical at closest approach, it is shown in [9] that the mass of the tether is approxi- mated by:

m =Prmo(mo+m,)Ay2 t %m,

where AV is the change in velocity required at the target planet in order to achieve capture.

In [9] a comparison is made between the mass of the tether [eqn (7)] and the propellant mass required

Radius

(Lm)

3550

3500 %

3450 P %

=4l-l-l 100 200 300 400

Time (s)

360, . , 2.4

a

0 -2.4

100 200 300 400 Tie (s)

Fig. 4. Orientation angle, a, and ai.

(to slow down the orbiter) with a chemical rocket system, Am :

Am = mo(e@@bn) - 1) (8)

where g is the acceleration due to gravity at the Earth’s surface and IsP is the specific impulse. The value for ZsP is assumed to be 300 s. The results of this comparison, presented in Table 1, are quite exciting because in every case the tether mass is smaller than the required propellant mass. The greatest absolute savings occur at Mars (144 kg) and the greatest percentage savings are found at Jupiter (81%). The design tension, computed by the simple formula[9]

T = mAm, + mnp) 4m,l

AP (9)

is a very good approximation for the actual tension observed in the simulation of the fly-through maneuver.

In every case, the design specifications (l)-(3) result in a tether length of approx. 1.8H (scale heights), so that the maximum atmospheric density at the probe is approx. 6 times (e” times) larger than that at the orbiter. The difference between the orbiter minimum height and the probe minimum height is nearly the full length of the tether, which reflects the fact that the tether is in a local vertical position, as desired, during the closest approach phase.

The specific characteristics of the Mars maneuver are shown in Figs 3-5. Figure 3 shows the radius of

Force 7500-

loo 200 300 400

Tie (s)

Fig. 3. Orbiter and probe trajectories. Fig. 5. Tension and normal forces on probe.

P w

Val

ues

Ven

us

Tab

le

1. A

eroc

aptu

re

resu

lts

for

sola

r sy

stem

ex

plor

atio

n (v

ertic

al

dum

bbel

l de

sign

)

Ear

th

Mar

s Ju

pite

r Sa

turn

U

ranu

s N

eptu

ne

Tita

n

AV

(km

/s)

0.35

0.

39

0.67

0.

27

0.41

0.

50

0.34

1.

31

Prop

ella

nt

mas

s (k

g)

126

142

256

96

149

185

122

559

Tet

her

mas

st

(kg)

31

.0

38.0

11

2 18

.0

42.0

63

.0

29.0

42

6 Sa

ving

s (%

) 75

%

73%

56

%

81%

72

%

66%

76

%

24%

Sa

ving

s (k

g)

95.0

10

4 14

4 78

.0

107

122

93.0

13

3

Len

gth

(km

) D

iam

eter

(m

m)

Pro

be

area

(m

*)

10.8

9.

0 14

.5

36. I

54

.4

1.42

1.

73

2.34

0.

600

0.74

0 99

9 81

8 60

5 23

70

1910

Des

ign

tens

ion

(N)

Act

ual

tens

ion

(N)

%

of

desi

gn

Orb

iter

pm.

(kg/

m’)

Pro

be

P,,.

(k

g/m

’)

Orb

iter

h,,,

(km

) P

robe

h,

,, (k

m)

5670

84

50

1550

0 10

10

1550

54

30

7850

12

700

1050

16

30

96%

93

%

82%

10

4%

105%

5.02

x

10-s

3.

04

x 10

-l

146

135

5.39

x

10-u

3.

27

x lo

--’

103

94

2.18

x

IO-’

4.

81

x IO

-”

1.73

x

10--

g 1.

42 x

W

6 2.

92

x 10

m9

1.05

x

10-s

93

49

5 84

8 81

45

9 79

4

72.7

72

.7

84.2

0.

780

0.53

0 I .8

9 18

10

2670

74

7

1720

79

5 10

100

I860

89

9 94

43

108%

11

3%

93%

3.29

x

IO-’

1.

31 x

10

-g

5.21

x

IO-’

2.

04

x 10

-s

8.14

x

IO-’

2.

28

x lO

-6

I496

12

46

522

1423

11

73

456

Atm

osph

ere

Scal

e he

ight

(k

m)

Ref

. al

titud

e (k

m)

Ref

. de

nsity

(k

g/m

’)

6 5

8 61

50

6458

35

07

1.0

x 10

-d

7.7

x 10

-e

5.5

x 10

-s

20

30

40

40

45

7159

2 60

350

2614

5 24

750

2738

1.

85 x

IO

-’

2.8

x 10

-2

4.69

x

IO-’

5.

76 x

IO

-’

1.7

x 10

-j

Hoh

man

n tr

ansf

er

Gra

vity

co

nsta

nt

(km

’/s2

) R

adiu

s (k

m)

V,

(km

/s)

App

roac

h e

App

roac

h rw

r

3.25

x

IO’

6.05

x

IO3

2.71

1.

14

6190

3.99

x 10

5 6.

38

x IO

’ 2.

97

1.14

4.28

x

IO4

3.40

x

10’

2.65

I .5

7 34

90

1.27

x

IO8

7.14

x

10’

5.64

I .0

2

3.79

x

IO’

6.00

x

10’

5.44

I.

05

5.80

x

106

6.85

x

10b

9.0

x IO

’ 2.

54

x IO

’ 2.

43

x IO

’ 2.

58

x IO

’ 4.

66

4.05

4.

01

1.10

I .0

6 6.

50

7190

0 60

800

2690

0 25

500

3080

tApp

roxi

mat

ed

from

eq

n (7

).

Aerobraking tethers 493

the orbiter, R,, and of the probe, R,, with respect to the center of the planet. Note that the radius of Mars is 3398 km. During atmospheric fly through the mini- mum altitudes of the orbiter and the probe are 92.5 and 80.7 km, respectively. The difference between these values is nearly the length of the tether. Next, the orientation of the tether, s( (with respect to the local vertical), and its spin rate, ai, are shown in Fig. 4. The graph clearly shows that during fly through the tether remains at a near vertical orientation (a = 0). The values of the normal and tension forces at the probe end of the tether are plotted in Fig. 5 (the forces on the orbiter are very similar). The graph shows that forces due to spin are equal before and after impact (due to the spin matching condition). Also, the normal force is close to zero, which was one of the goals of the design. Note that the tether designed for this maneuver has an ultimate strength of 15,500 N [from eqn (9)] which is higher than the maximum value of tension observed in the simu- lation Also note that the actual tension due to spin is, in this case, overestimated by eqn (9).

The promising results obtained in [9] and summar- ized in Table 1 for the rigid rod model demand that a deeper examination of the potential of aerobraking tethers be made. In the following analysis, we examine the question of optimal tether mass.

3. OPTIMAL TETHER MASS MANEUVERS

In the analysis performed in [9], the design process is structured to develop an aerobraking maneuver that requires a low tether mass (i.e. a low tether strength) in order to compete favorably with the propulsive systems. However, the starting point in the analysis assumes a dumbbell type maneuver, with the tether in a nearly vertical orientation at periapsis. In [8], a vast range of maneuvers seems possible between the extremes represented by the (vertical) dumbbell and (horizontal) drag chute maneuvers, and the possibility that some of these maneuvers may result in lower tether masses remains unexplored. One of the reasons to address the dumbbell case first is the ease with which the normal forces can be analyzed and the system designed to eliminate them. This requirement arises from the fact that the rigid rod model lacks the sophistication to incorporate the bending associated with normal forces. The develop- ment of the flexible models in [lo] and [1 1] validates the rigid rod as an excellent design tool and also allows the analysis of a wider variety of tethered systems. At this point, a more systematic analysis of the possible tether aerobraking maneuvers is not only possible but also very desirable.

Before we proceed in the next section with the optimization problem, we should mention briefly that optimization problems, in general, can take many iterations to converge and can be very sensitive to computational round-off and/or truncation errors. This is especially true for most practical problems-

like the one treated in the present paper-where no explicit formulae for the gradient or the Hessian of the cost function are available. One must resort to numerical calculation of these quantities using finite difference methods, thus introducing an additional error in the calculations. Most optimization algor- ithms therefore benefit from good starting guesses for the solution. A good starting guess, apart from the fact that it can accelerate the convergence of the algorithm, can lead to the global rather than to a local minimum. One therefore has to be very careful in choosing initial guesses. Experience, judgement and (sometimes) luck can be the difference between success (convergence) and failure (divergence). There- fore, very complex problems are best solved by an evolutionary approach, whereby a problem with a smaller number of independent variables is solved first. Solutions from lower order problems can then be used as starting points for more complex problems with more independent parameters.

With this philosophy in mind we first solve the minimum force problem for an aerobraking tether. In this formulation the length of the tether is kept fixed, and the optimization algorithm will provide the mini- mum force maneuver for the given tether. One can then redesign the tether (e.g. change its diameter) to withstand the required force by the minimum amount of mass. Therefore, the minimum force solution for a given length of the tether is dual to the problem of minimum mass for this tether length. The results of the minimum force problem can then be used, as mentioned earlier, as good initial guesses for the complete minimum mass problem, where the length of the tether is among the design parameters.

3.1. Minimum force analysis

We initiate the search for the optimal tether maneuver by determining the aerobraking maneuver that minimizes the forces on the tether. For a given design, this optimum maneuver can be analyzed as a constrained optimization problem and solved using non-linear programming techniques. Once the physi- cal parameters for the system are specified, the tether aerobraking maneuver can be defined by three parameters: the initial conditions outside the atmos- phere, or, and oi,, and the radius of periapsis for the hyperbolic orbit, rper . Thus the optimization process involves the search for the optimum initial conditions in a three dimensional parameter space. For the analysis presented here, the total force on the tether is defined as the (magnitude of the) vector addition of the tension and normal forces. Note that, for a given set of initial conditions, determination of the maxi- mum force requires the computation of the complete atmospheric fly-through maneuver. Therefore the maximum force is computed only a posteriori. Clearly, an analytic computation of the gradients of the cost (i.e. force) with respect to the design par- ameters is out of the question. In addition, some constraints must be placed on the problem in order

494 J. M. Longuski et a/

to assure that the resulting optimum trajectory provides an acceptable aerobraking maneuver. First, we provide the value of the final orbit eccentricity that must be achieved by the maneuver, e,, and introduce it as an equality constraint. Next, the objective of keeping the orbiter higher than the probe is introduced as an inequality constraint on the difference in the minimum altitudes of the orbiter and the probe, Ah,, during the maneuver. Finally, the presence of compressive forces on the tether is prevented with an additional inequality constraint. Mathematically the problem can be written as:

minimize: F,,,(x)

x = ]&I 5 &I 9 rperlf

subject to: e - e, = 0

Ah -Ah,>0

T,,, ’ 0

(10)

(11)

(12)

(13)

where e and Ah are the eccentricity and altitude difference achieved by the maneuver and T,, is the minimum tension on the tether.

A large variety of techniques exists to solve a problem of the form given above. In this paper optimization software based on the exterior penalty method[l3,14] is applied to the tether problem. This method incorporates the active constraints in the objective function by performing successive uncon- strained minimizations of the function:

Fk=F,,,(x)+r$G,(x), k=1,2,3 ,... ,=I

and increasing the value of the factor, rkr after each iteration. Here G,(x) is some function of the con- straints g, (j = 1, 2, 3) and r, is a positive constant known as the penalty parameter. For our analysis we choose

G,(x) = {max]O, g,(x)1)2

The unconstrained minimizations are performed using Powell’s method with a golden section method for the one-dimensional minimizations [13,14]. In the optimization procedure, the rigid rod model is the only practical model due to the large number of iterations required. (Typically the force optimization process requires several hours of CPU time on a Sun Spare 10 workstation using the rigid rod model.) The validity of the optimum maneuver can be determined by performing a flexible analysis after the optimiz- ation process, as in [I 51.

The optimization method is first applied to the Mars aerocapture design [9,15]. The parameters are taken directly from the results of that maneuver (see Table 1 where e = 0.9999 and Ah = 12 km). The resulting optimum maneuver (see Figs 6 and 7) is significantly different from the maneuver described in Figs 4 and 5. During the optimal fly through, the minimum orientation angle of the tether is greater

270 , 225 f& =12 180 = 10

sa

aMeg) 135

90

45 0

7

Oil--n-i 100150200250300350400

Tie (s)

Fig. 6. Orientation angle (optimum force trajectories)

than zero. This fly-through angle means that the tether is subjected to significant tension at all points in the trajectory, which reduces the tendency to bend (in a flexible tether). In addition to these favorable characteristics, the maneuver reduces the forces from over 12,000 N (Table 1) to 9000 N while achieving the same eccentricity. (Note in Fig. 7 the large reduction over Fig. 5 in the forces before and after impact due to the reduction in spin rate.) Small normal forces appear in the maneuver since the effective area of the tether is reduced by the fly-through orientation and this slightly offsets the aeromatching condition. Also note that the spin-matching condition emerges, in a natural way, as part of the optimum solution.

We observe that the altitude constraint is active in the previous maneuver (Ah, = 12 km), and wish to investigate the effect of a reduction of the A& require- ment. When this is done the forces are further reduced as the Ah, requirement is relaxed, until a value of 8 km is reached. Beyond this point, any further reductions in Ah, have no effect on the maneuver (see Figs 6 and 7). Thus, the optimum maneuver for an unconstrained clearance maintains a clearance of 8 km. In this maneuver, the forces are reduced to the minimum value of about 7000 N, with a fly-through orientation of about 55. at closest approach.

The results of further investigation into the mini- mum force problem with unconstrained altitude are presented in Table 2 for the Mars tether with different

mo 8alo 7ooo 6000

Force 5OL-NI

04000 3ooo 2000 lOCKI

0

loo 200 300 4& So0 Time (s)

Fig. 7. Forces on the probe (optimum force trajectories).

Aerobraking tethers 495

Table 2. Mars minimum force results for unconstrained altitude7

e A V (km/s) urn,” (degrees) Ah F,, (N)

0.50 1.33 64.0 (72.5) 3.82 14,400 (I 3,200) 0.75 0.983 62.2 (67.7) 5.65 10,700 (9720) 0.99 0.676 55.3 (60.2) 8.21 6990 (6620) 1.20 0.422 39.2 (48.4) 11.3 3990 (3970) 1.40 0.192 30.1 (28.0) 12.5 1260(1370) 1.50 0.0819 -0.660(17.1) 14.5 249 (356)

tValues in parentheses were predicted from eqn (47).

target eccentricities. In Fig. 8 we plot the maximum tether force, F,,,,,, and the minimum orientation angle, amin, against A.V. We see that F,,,, increases nearly linearly with AV while Q, asymptotically approaches 90”, which is equivalent to a drag chute maneuver. For low AV, the u,,,~” is nearly zero, approaching the vertical dumbbell maneuver. We will refer to this new type of maneuver (i.e. a maneuver having non-zero tl,in) as an inclined maneuver.

3.2. Sliding pendulum model

During minimum force fly through we notice that near the time of closest approach, the tension force remains nearly constant, when the altitude is uncon- strained. This is evident in the plateau that appears in Fig. 7 for Ah, < 8. Another interesting feature that can be observed is that the orbiter altitude remains nearly constant during the atmospheric fly through, while the probe swings down into the atmosphere. This behavior is apparent in both the vertical dumb- bell maneuver (Fig. 3) as well as in the optimal force and optimal mass solutions[lS]. The pendulum motion of the probe suggests that the or- biter-probe-tether system can be modeled as a sliding pendulum, where the orbiter represents a sliding attachment point and the probe and tether represent the pendulum (see Fig. 9). In this motion the probe swings down to a minimum orientation angle, a,,,, and is bounced back up and out of the atmosphere, while the orbiter maintains a nearly constant altitude. With this behavior in mind, we will attempt to construct a simple analytic model of the tension force in order to better understand the dynamics of the problem and to guide the numerical determination of the minimum force and minimum mass trajectories.

The A I/ obtained from the atmospheric fly through

0 0.20.40.60.8 1 1.2 1.4

AV Ws)

Fig. 8. Minimum force results with no altitude constraint (Mars).

can be approximated by an impulse analysis for a particle[l6,17]:

AV=rprVWr

(To find the impulsive AV for the entire tether see [8]. For a more sophisticated analysis of atmospheric fly through see Vinh et al. [18] and Vinh [19]). Equation (14) is given in terms of periapsis conditions, as indicated by the subscript “per”, so that rpcr is the radius at periapsis, V,, is the velocity at petiapsis and ppr is the atmospheric density at periapsis. The drag coefficient, Co, corresponds to a cross sectional area of A. The variable e is the approach eccentricity, relative to the planet or satellite. The term /3 is the inverse scale height (/I = l/H). (It is interesting to note that, according to Vinh [19], the product fir,, is nearly constant for variable rper and /I.)

Since the drag at periapsis is given by

we can write, from eqn (14), that

(15)

mAV fp,,C,,A = -

rper Vper (16)

so that the drag can be rewritten as

Dper= mAV (17)

Noting that the velocity at periapsis can be expressed in terms of the local circular speed, V,,, we have

VP,,= J

+(l +e)=&GV,, (18) per

and we obtain

D,,=mAV (1%

The expressions for drag [eqn (19)] and A V [eqn (14)] imply that a characteristic fly-through time can be computed as

(20)

Fig. 9. Sliding pendulum.

AA 3X-B

496 J. M. Longuski et a/

which turns out to be about 95 s for Mars aerocap- ture. (Note that this value is very close to the actual fly-through times observed in the simulations in Figs 3-7.)

We can now derive the equations of motion for the sliding pendulum depicted in Fig. 9. Here we assume that the drag force, D, acts only on the probe. The orbiter is assumed to slide along a “frictionless wire” which provides a vertical reaction force, R. In this sliding pendulum there is no gravity. (The wire is, of course, a fictitious device which represents the hori- zontal trajectory of the orbiter). There are two de- grees of freedom represented by the variables x and cc; the unit vectors i,i, i? are fixed in the inertia1 reference frame.

The location of the center of mass of the tether system is given by

r=(x-I,sinol)Z+I,coscr3 (21)

where f, is the distance from the orbiter to the center of mass:

(22)

and where m is the total mass,

m =m,+m,+m, (23)

Differentiating eqn (21) twice, we obtain

f = (c? - l,di cos CI + /,oi’sin cr)T

-(Z,&sincr +I,oi2coscc)~ (24)

Thus, from Newton’s second law we have

D R-10dicosu+l,ci’sinu=-- (25)

m

I,disincr +l,ci2cosc1 =R (26) m

To remove the unknown, R, from eqn (26) we can use Euler’s law

M’=jl’ (27)

The system angular momentum about the center of mass is

H’=loifc (28)

where I is given by [ 151:

m, (C + 1:) I =m,Ii+ 3, + m,li (29)

and where

, =l(m,+jm,) P m

I = I, + I, (30)

The moment about the center of mass is

M’=(I,Dcoscc -L,Rsina)k (31)

Substituting eqn (31) and the derivative of eqn (28) into eqn (27) provides

Iti=I,Dcosa -1,Rsina (32)

Eliminating R from eqns (32) and (26) we obtain

$= I,Dcosu -ml~ci2cosu since

I + ml2 sin2 a: (33)

0

Equations (25) and (33) provide the final equations of motion. Our main goal in deriving these equations is to provide an expression for the tension in the tether. This is most easily obtained by considering Newton’s law for the orbiter, which moves along the x direction and is subject to a single force component along x:

m,i = -Tsincr

Thus, the tension is .

T=__m,x an tl

Substituting for f from eqn (25) into eqn (35), we obtain

m,D T=-- m,l,di cos a

m sin u sin c( + m,10ci2 (36)

Next, we can eliminate & in eqn (36) through expression (33) to obtain our final relation for the tension:

Dm T=” l-

m1,1, cos’ u

m sina I + ml: sin2 u >

+m,10ci2 ( 1 + ml: cos2 u

I + ml: sin2 u > (37)

Our hypothesis is that the minimum force maneu- ver maintains a nearly constant tether tension during fly through, so that if we can calculate the value of T at two points in the atmosphere and set the two values equal, then we may find an approximate value for the optimal amin (given in Table 2 for Mars).

The tension comes from two sources: spin tension and drag tension. Naturally, before the tether enters the atmosphere it is subjected to only spin tension, but upon entry, drag tension becomes apparent. Of course as the tether translates through the atmos- phere the moment from the drag force reduces the spin rate and hence the spin tension. When the spin rate becomes zero

ci=o (38)

the system reaches its minimum orientation, u,,,, and eqn (37) becomes

D m T(u,,,)=+

m1,1, cos? a,,,

I + ml: sin’ 5(,,, > (39)

m sin amin

At an earlier stage, as the tether just enters the sensible atmosphere, a suitable value for the entry a can be chosen, u,,,~~, where the spin rate ci,,,,, has not been significantly altered and where drag tension is an

Aerobraking tethers 497

important effect. Following the approach of [9] and must be included in the analysis. However, since we

[15], we can determine the change in spin rate from want the tether to be only strong enough to withstand the law of angular impulse and angular momentum. the forces in a particular maneuver, we can eliminate We assume that the linear impulse, P, is applied at the the diameter from the optimization space. The mini- probe to cause the required change in velocity, AV: mization can then be defined as:

P=(m,+m,+m,)AV (40)

At the time of closest approach, the orientation of the tether is tilted by an angle a,in. The angular impulse at this orientation is given by

PI, cos a,in = IAQ (41)

where the reference point is the center of mass, AR is the change in inertial spin rate and IAR is, of course, the change in angular momentum.

From eqns (41), (40) and (30) we find that the change in spin rate is

A~=(mO+~m~)~A~cosa, I mm (42)

Keeping in mind the spin matching condition (1) and that d can be approximated by the inertial spin rate, R, we have

minimize: m,(x)

x = [aO, ai,, rpcr, 4 (48)

subject to: e - e, = 0 (49)

Ah -Ah,>0 (50)

T,,, > 0 (51)

where m, is the tether mass. (Note that the compu- tation of m,, for a given vector x, includes the numerical (iterative) determination of the tether diameter required by the forces due to that particular maneuver. This, along with the inclusion of the length as a fourth dimension in x, increases the number of iterations, and increases the required CPU time significantly.)

c?,,,,, z Qi, = -tAQ (43)

At a,,,,), we can determine the tension from drag by using the exponential atmosphere relation

D enlry = D,, exr4 - @OS a,in - ~0s a,,,,Yf4 (44

so that the total tension is found by substituting eqns (43) and (44) into eqn (37). The only remaining issue is to estimate the value of acntry. Heuristically, we find that

Before proceeding to the numerical computation of optimal tether masses for the various solar system bodies, we first consider an analytic approach which may be able to predict the characteristics of the optimal mass solution. We have already seen in the minimum force analysis that the minimum force often occurs at non-zero ci,,,. This leads to the possibility that a minimum mass solution may exist at some non-zero a,,, .

x,,t,y x 90” (45)

Intuitively, eqn (45) seems reasonable because at 90 no appreciable spin rate change can occur, and yet the effect of drag must be noticeable since at this point the probe is only one tether length above its lowest altitude in the atmosphere. For the tether designs given in Table 1, this altitude is only 1.8H, which will provide some drag tension. Thus, our expression from eqn (37) becomes

We also observe that another minimum mass candidate solution is the vertical dumbbell maneuver. This solution has the great advantage of being easy to calculate from eqn (7). Thus we have two candi- date solutions for minimum mass: (i) the vertical dumbbell maneuver (a,,, = 0) and (ii) the inclined maneuver (amin > 0).

We can find an approximate analytic solution for the inclined maneuver by the following procedure. First we take the characteristics of the vertical dumb- bell tether given in Table 1 (AV, m,, I, T, H, V,, , rperr e) where all the values are calculated analytically. [For example we find T from eqn (9) we estimate rper to be several scale heights above the planetary radius, and we compute VIc from & .] We assume that the clearance altitude is given by Ah, = /,,_ (where 1 des,gn is the design length given in Table I). Next we substitute these characteristic design values into eqn (47) and solve for a,,,,“. Naturally the resulting value will be non-zero which means that the clearance requirement (Ah > Ah,) will no longer be satisfied. Thus we must correct our estimate of the tether length, I, by the following expression

(mo+‘mt)‘AVcosa 21 ml” 1 * (46)

We can now equate relations (39) and (46) to solve for a,,,:

f(c,,” 1 = T(aemry I- T(amin) = 0 (47)

Table 2 shows the results of eqn (47) in parentheses. These are in surprisingly close agreement with the optimal force solutions.

3.3. Minimum mass analysis

In order to analyze the tether mass problem, the dimensions of the tether, diameter d and length I,

I = Ah,/cos(a,,,) (52)

In the next step we substitute the new value of I into eqn (47) to find a new a,,,. After a few iterations (which can be performed on a hand calculator) we converge to a solution which provides an amin that satisfies the minimum force condition (47) and the

498 J. M. Longuski ef al.

Table 3. Analytic prediction for optimal tether mass

Vertical dumbbell solution (qn = 0) Inclined solution (am,” > 0)

Body m, (kg)

Venus 31.0

T (N)

5670

1 (km)

10.8

m, (kg)

26.9

T(N)

4500

1 (km)

12.6

amln (degrees)

31.0 Earth 38.0 8450 9.0 32.8 6070 10.8 33.7 MUX 112.0 15500 14.5 67.4 6570 20.6 45.2 Jupiter IS.0 1010 36. I 28.5 I560 37. I 13.5 Saturn 42.0 1550 54.4 45.6 1790 57.9 20.0 Uranus 63.0 1720 72.7 72.9 I840 78.7 22.5 Neptune 29.0 795 72.7 40.8 1070 75.7 16.1 Titan 426.0 10100 84.2 298.0 5330 112.0 41 .o

altitude clearance (52). However, this solution still assumes the original design mass mdaign (from Table 1) which is no longer correct because the length and the strength requirement have changed. We can compute a new estimate of the tether mass from

Y = (~/L&n) [T(a,i”)/TdesignlI?ldcsign (53)

where “design” refers to the values in Table 1. Substituting the new estimate of m, into eqn (47), and repeating the above procedure, we converge to new values for I and a,,,. If the values change signifi- cantly, we can repeat the process with a more refined estimate of m, from (53).

The results of the iterative procedure are presented in Table 3 under the heading “inclined solution”, where we used the precise value for rPCr in Table 1 (which was obtained from simulation). For con- venient comparison we also present the analytical results from the vertical dumbbell solution. The predictions of Table 3 from the analytic approach are quite interesting. According to the table, the optimal mass maneuver for Venus, Earth, Mars and Titan is an inclined maneuver, while the vertical dumbbell maneuver is expected to be optimal for Jupiter, Saturn, Uranus and Neptune. In the cases of Venus, Saturn, Earth and Uranus the differences between the two candidate maneuvers are probably too small to be decisive, and await numerical optimization for the final judgement. On the other hand, the case of Jupiter clearly indicates a vertical dumbbell maneuver while the case of Mars clearly favors the inclined maneuver.

350 300

250

200 a&g) Iso

100 50

a

v

0

0 100 200 300 400 500 Tii (s)

Fig. 10. Orientation angle (optimum mass trajectory).

Radius (km)

3650 e

3600

3550

3500

IQ

%

3450 1 1.6

1.5

1.4

1.3

1.2

1.1

1

Ecc

34oodo.9

0 100 u)o 300 400 500 Tie (s)

Fig. 11. End point positions and eccentricity (optimum mass trajectory).

Since the inclined case of Mars provokes some interest, we will first consider searching for its mini- mum mass by the numerical optimization techniques discussed earlier. In these computations we assume that Ah, = 14.5 km and e, = 0.9999. The resulting optimum tether system has a length of 20.7 km (an increase) and a diameter of 1.51 mm. These par- ameters represent a tether mass of only 66.4 kg, a saving of almost 41% over the tether system in Table 1 and 74% over propellant mass. We note that the close agreement of the optimal result with the ana- lytic result presented in Table 3 is rather remarkable. In addition, the resulting inclined maneuver (Figs 10-12) presents some clear advantages over the verti- cal dumbbell maneuver. First, the spin rates before and after the maneuver are much smaller, eliminating the need for large fuel expenditures to generate spin rate. The fly-through attitude of about 45” maintains

0 100 200 300 400 500 Tie (s)

Fig. 12. Tether forces on the probe (optimum mass trajectory).

Aerobraking tethers 499

Values

AY(km/s)

Table 4. Aerocapture results for solar system exploration (optimal mass)

Venus Earth Mars Jupiter Saturn Uranus

0.35 0.39 0.67 0.27 0.41 0.50

Neptune Titan

0.34 1.31

Propellant mass (kg) 126 142 256 96 149 185 122 Tether mass (kg) 25.9 30.5 66.4 18.8 44.1 67.2 32.6

Savings (%) 79% 79% 74% 80% 70% 64% 13% Savings (kg) 99.1 112 190 11.2 I05 118 89

Length (km) 12.4 IO.5 20.7 36.1 54.4 72.7 72.8 Diameter (mm) I .22 1.43 I.51 0.607 0.757 0.809 0.563 Probe area (m*) 999 818 605 2370 1910 1810 2670

Maximum force (N) 4180 Minimum tl (degrees) 27.1 Maneuver type? I

Orbiter h,,,(km) 145 103 Probe h,,,(km) 135 94

5820 6420 28.4 45

I

0.46 V

I850 0.27 V

95 495 848 1496 1246 540 80 459 794 1423 II73 456

1050 1630 0.18 V

899 0.75 V

559 282

50% 271

112 1.33 747

5030 41

tl =inclined, V = vertical.

significant (drag) tension on the tether throughout the maneuver (see Fig. IO) which should prevent it from bending despite the fact that the changes in tether dimensions create a system that (slightly) violates the center matching and aeromatching conditions. It is interesting to note that, even though it was never introduced as a requirement, spin matching is a feature of the optimum trajectory, as indicated by the equal tensions before and after fly through.

Table 4 presents the optimal mass for aerobraking tethers for solar system exploration. Again the orbiter and probe masses are assumed to be 1000 kg each, the tether is constructed of Hercules AS4 graphite and the capture orbit relative to the planet is assumed to have an eccentricity of 0.9999. phese are conditions (4)-(6) of the earlier tether design]. In addition we retain the probe areas of Table 1 and set Ah, equal to the tether lengths in Table I, which are given (to close approximation) by

Ah, z 1.8H (54)

The results of Table 4 show surprising agreement with the analytical predictions of Table 3. The ana- lytical approach provides similar values for mass and length to the optimal solution and, furthermore, correctly determines whether the optimal maneuver is vertical or inclined.

One word of caution should be mentioned about checking the convergence of the numerical algorithm for the optimization problem. In all the available constrained optimization algorithms in the literature, the identification of the optimum solution is very important, from the points of view of stopping the iterative process and using the solution with confi- dence[l3]. Generally speaking, one has two ways of checking for, at least, a local minimum. The first is by comparison with neighboring extremals (by per- turbing the design vector near the candidate opti- mum, for example). Any such extremal should give a larger value of the final cost, or else, a lower value of the cost should be obtained only at the expense of violating one (or more) of the constraints. The second is by testing the Kuhn-Tucker conditions[l4]. The first condition was checked for all the previous cases

(the Kuhn-Tucker conditions were checked period- ically) and were found to be satisfied (within a numerical error), providing at least a local minimum. The question of whether global minima can be found remains open at this stage. However, preliminary results presented in [20] provide a strong case that the minima given in Table 4 are, in fact, global.

4. CONCLUSIONS

The analytic approach presented here is remark- ably accurate in predicting which of the two aero- braking strategies (vertical or inclined) will provide a lower tether mass. It is not yet known if the resulting solution is a global minimum. However, both maneu- vers provide significant mass savings over propellant, suggesting a new option in the development of mis-

sions to the atmosphere-bearing planets.

Acknowledgemen(s-The authors thank John Mechalas for assistance in computational work. This work has been supported in part by the National Science Foundation under Grant No. MSS-9114388.

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